Course:Harris, Fall 08: Diary Week 4

Mon:

• I handed out copies of a color outline on classifying Wallpaper Groups (from [1]).
• Groups finished up the Frieze Patterns Exploration (took most of the period).
• Question raised: If a frieze has reflection symmetries from both a horizontal line and from vertical lines, must it also have a 2-fold rotation symmetry?
  • The list of Frieze Groups suggests this to be the case, as there is only one Group which contains both horizontal and vertical symmetries, and it also contains rotational symmetries; but is this truly the case?
  • Suppose a frieze design has these symmetries:
    • from reflection across a horizontal line; call that transformation H.
    • from reflection across a vertical line; call that transformation V.
  • We know that if we apply either H or V to the plane, that the frieze design is moved onto itself; but is there a rotation which moves the design onto itself?
  • We examined what happens to the plane if we apply, successively, first the transformation V and then the transformation H to it:
    • Tracking where labeled points of the plane get sent by doing V and then H to the plane, we saw the net result is a 180-degree rotation:
      • Center point is the intersection of the vertical and horizontal lines.
  • Since the design gets preserved by both the the V and H transformations, doing both of those must also preserve the design; that means that the design is preserved by the rotation, as well. In other words:
    • Yes, having a vertical-line symmetry and a horizontal-line symmetry in a design implies the design has a 2-fold rotation symmetry, also.
• The general goal of the course is to investigate the mathematics behind design symmetries.
  • Part of this is simply to learn to classify the various groups of symmetries that can occur.
  • Another part is to see how various symmetries interact with one another, as in the analysis above.
• Wednesday will be a quiz on building designs, using a given motif and a given Rosette or Frieze Group .
  • You can use the Frieze Group outline to assist you; but you should know the Rosette Groups by now.

Wed:

• I handed back corrections to Exercises and such, noting that "cyclic" is not a group-name, but C4 (for instance) is.
• We spent about 10 minutes on Quiz 1 (make designs, using given motif, with symmetry groups of C4, D4, M1, and 12).
  • Most trouble seemed to be with D4, ensuring that the design does, indeed, have 4-fold rotational symmetry. Need to double the given motif in order to have a reflection-symmetric building block.
• Problems in how to specify a tessellation of the plane by, say, copies of one parallelogram (20 minutes or so):
  • Issue is in showing that one knows how to cover the entire plane. Does saying "and so on" really nail it down?
  • If we have rows of parallelograms which are simply related to one another, then we can say "and so on" and have that mean something definite.
    • This is a tessellation with symmetries. This is one reason symmetries are useful, for uniquely specifying a tessellation of the entire plane (infinite in extent!).
  • If rows are randomly related to one another, then there is no unique specification.
    • This is a tessellation--or, rather, a class of many possible tessellations--without symmetry.
• What is a lattice of translations on a tessellation (at least, on a Wallpaper-group tessellation, i.e., on that has at least two translations in its symmetry group)?
  • Pick any point, anywhere.
  • Note the position of at images of that point under translations.
  • Note positions of images of those images under translations, and so on.
  • Connect these images by lines that go in the directions of the translations; these are the lattice lines associated to the original choice of a point.
• We had 10 or 15 minutes to start on the Wallpaper Group Exploration; to be continued next time.

Fri:

• Most did fairly well on the quiz.
• I emphasized that Exercises should be corrected to get a decent grade; I am available for help with them.
  • Particularly troublesome in the Frieze Exercises was finding all the symmetries in a frieze.
  • Note that there can be "trivial" glide reflections when there is a horizontal reflection present (trivial because the separate translations and reflections are both symmetries); **but in other cases, there is a nontrivial glide reflection, i.e., in a frieze which does not have a separate horizontal reflection.
• I changed the wording of #7 in Frieze Exercises to be restricted to the case of a nontrivial glide refelection.
• Groups spent most of the period (20-30 minutes) finishing up Wallpaper Symmetry Exploration.
  • I asked the class to spot the nontrivial glide reflection in Schattschneider #91 [2].
    • This has, for its glide reflection axis, the line along the bugs' legs.
• Groups spent a short while getting started on Escher's Wallpaper Groups Exploration.